Question
Construct a triangle ABC with AB = 5 cm, BC = 6 cm, and CA = 7 cm. Draw the circumcircle of this triangle.
Solution — Step by Step
Draw a line segment BC = 6 cm. Label the endpoints B and C. This is your base.
Use a ruler; mark B on the left and C on the right. The accuracy of your construction depends on this first step — measure carefully.
From B: Set compass to 5 cm. Place compass at B and draw an arc above the line BC.
From C: Set compass to 7 cm. Place compass at C and draw an arc that intersects the arc from B.
The intersection point of the two arcs is vertex A.
Connect A to B and A to C. Triangle ABC is complete with AB = 5 cm, BC = 6 cm, CA = 7 cm.
The circumcircle passes through all three vertices A, B, and C. Its centre (called the circumcentre) is equidistant from all three vertices.
The locus of all points equidistant from two points B and A is the perpendicular bisector of BA. So the circumcentre lies on the perpendicular bisector of every side.
To find the circumcentre: draw perpendicular bisectors of any two sides — their intersection is the circumcentre.
Perpendicular bisector of BC:
- Set compass to more than half of BC (e.g., about 4 cm).
- Place compass at B; draw arcs above and below BC.
- Without changing compass setting, place at C; draw arcs cutting the first arcs.
- Join the intersection points. This line is the perpendicular bisector of BC.
Perpendicular bisector of AB (or AC):
- Repeat the same process for side AB.
The two perpendicular bisectors intersect at point O — this is the circumcentre.
Set compass to distance OA (= OB = OC, the circumradius). Place compass at O. Draw a circle.
This circle should pass through all three vertices A, B, and C. If it doesn’t, recheck your perpendicular bisectors.
Verify: Measure OA, OB, OC — all should be equal (within 1 mm of each other for a well-drawn construction).
Why This Works
The circumcentre is equidistant from all three vertices because it lies on the perpendicular bisector of each side. The perpendicular bisector of a segment is the set of all points equidistant from both endpoints. So:
- O on perp. bisector of BC → OB = OC
- O on perp. bisector of AB → OA = OB
- Together: OA = OB = OC = circumradius
For a scalene triangle like (5, 6, 7), the circumcentre lies inside the triangle because all angles are acute. For an obtuse triangle, the circumcentre lies outside. For a right triangle, it lies at the midpoint of the hypotenuse.
Alternative Method — Calculating Circumradius
For verification, calculate using the formula:
where are sides and is the area.
For a = 5, b = 6, c = 7:
Using Heron’s formula:
Measure OA on your construction — it should be approximately 3.6 cm.
CBSE construction questions carry 4 marks. The marks are split as: drawing triangle (2 marks) + perpendicular bisectors and circumcircle (2 marks). Show all construction arcs — never erase them. Examiners check for arcs to confirm you used a compass, not a protractor/guessing.
Common Mistake
Students often draw only one perpendicular bisector, then estimate the circumcentre by eye — or they draw both bisectors but fail to extend them to where they meet. Both perp. bisectors must be drawn long enough to intersect clearly. Always extend them beyond the triangle if needed.